Weighted Generalized Risk Measure and Risk Quandrangle: Characterization, Optimization and Application

2026-03-11

AI Summary

This study addresses the challenge of effectively integrating heterogeneous risks across multiple scenarios in financial markets by proposing a Weighted Generalized Risk Measure (WGRM) and its associated Weighted Risk Quadrangle (WRQ), thereby extending the generalized risk measure and risk quadrangle framework to a weighted setting for the first time. Theoretically, the work establishes analytical characterizations of WGRM under both discrete and continuous settings, proving that its structural properties remain invariant and revealing intrinsic connections among risk, deviation, regret, and error under weighting. Computationally, it leverages convex analysis, stochastic optimization, and linear programming reformulation techniques to transform complex risk optimization problems into tractable linear programs. Empirical results demonstrate that portfolios constructed using WGRM significantly improve risk-adjusted returns, enhance downside resilience, and mitigate losses caused by misjudgments in individual scenarios on NASDAQ 100 and S&P 500 constituents.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Various financial market scenarios may cause heterogeneous risk assessments among analysts, which motivates the usage of the Generalized Risk Measure in Fadina et al. (2024, Finance and Stochastics). Effectively synthesizing these diverse assessments avoids over-relying on a single, potentially flawed or conservative forecast and promotes more robust decision-making. Motivated by this, we establish analytical characterizations of the Weighted Generalized Risk Measure (WGRM) under both discrete and continuous settings. Building upon the WGRM, we incorporate the Fundamental Risk Quadrangle (FRQ) in Rockafellar and Uryasev (2013, Surveys in Operations Research and Management Science) into the Weighted Risk Quadrangle (WRQ) and show that the intrinsic relationships among risk, deviation, regret, error, and statistics in FRQ are preserved under weighted aggregation across scenarios. Moreover, we demonstrate that certain complex risk optimization problems under the WGRM can be reformulated as tractable linear programs through the WRQ structure, thus ensuring computational feasibility. Finally, the WGRM and WRQ framework is applied to empirical analyses using constituents of the NASDAQ 100 and S&P 500 indices across recession and expansion regimes, which validates that WGRM-based portfolios exhibit superior risk-adjusted performance and enhanced downside resilience and effectively mitigate losses arising from erroneous single-scenario judgments.

Problem

• Generalized Risk Measure
• Risk Aggregation
• Heterogeneous Risk Assessments
• Weighted Risk Quadrangle
• Robust Decision-Making

Innovation

• Weighted Generalized Risk Measure
• Risk Quadrangle
• Heterogeneous Risk Assessment
• Tractable Optimization
• Downside Resilience